2002 Dodge Neon fuse/battery issue

Go to an Autozone or CSK/O'Reilly and ask them to check your electrical system. Battery icon turn on while the engine is running means that the electrical system is running at a low level, less than 12.6 volts. It should be between 13.5 to 14.9 volts.

I suspect your alternator, but before we settle with that, I want to see the electrical test. While you are testing that, have them check the physical condition of the battery (is it corroded, are the post tight, is there any surface discharged detected, etc) and if the tension of the serpentine belts is normal.